Attached Tip Mass Research Articles

Nonlinear dynamic analysis of a spring coupled double beam based piezoelectric energy harvester under parametric base excitation

This paper proposes a base excited, spring coupled double cantilever beam with attached tip mass based piezoelectric energy harvester. The dimensions of the beams are so chosen that the system experiences 1:1 internal resonance. The governing coupled electro-mechanical equations of motion of the system are obtained by using the Lagrange principle which is reduced to the temporal form by using generalized Galerkin’s method. The governing temporal equations of motion are in the form of that of a parametrically excited coupled system with geometric and inertial nonlinearities. These nonlinear equations are solved using the method of multiple scales and the results are compared with those solved by numerical method (4th order Runge Kutta method) for principal parametric resonance conditions. Parametric studies are conducted by varying the tip masses of the beams, coupled spring stiffness, load resistance, external excitation frequency and excitation amplitude. Results show that by introducing spring coupling, the system operational frequency range increases significantly and also energy transfer process takes place due to the internal resonance. Significant enhancement of voltage and power are observed in the present work in comparison to the earlier studied equivalent single beam-based energy harvester.

Revisited on the Free Vibration of a Cantilever Beam with an Asymmetrically Attached Tip Mass

Cantilever with an asymmetrically attached tip mass arises in many engineering applications. Both the traditional method of separation of variables and the method of Laplace transform are employed in the present paper to solve the eigenvalue problem of the free vibration of such structures, and the effect of the eccentric distance along the vertical direction and the length direction of the tip mass is considered here. For the traditional method of separation of variables, tip mass only affects to the boundary conditions, and the eigenvalue problem of the free vibration is solved based on the nonhomogeneous boundary conditions. For the method of Laplace transform, the effect of the tip mass is introduced in the governing equation with the Dirac function, and the eigenvalue problem then can be solved through Laplace transform with homogeneous boundary conditions. The computed results with these two methods are compared well with the numerical solution obtained by finite element method and approximate analytical solutions, and the effect of tip mass dimensions on the natural frequencies and corresponding mode shapes is also given.

Analytical model and energy harvesting analysis of a vibrating slender rod with added tip mass in three-dimensional space

In the paper, a new 3D energy harvesting system is provided. This work discussed the Lagrange approach to derive the differential equations of motion in the case of energy harvesting systems. An electromechanical system consists of a mechanical resonator, a piezoelectric transducer and electrical circuit with the load resistor. A flexible slender rod clamped at the bottom and loaded by the tip mass is proposed as the resonator. Moving in the 3D space, it enables the system to avoid the gravitational potential barrier of the straight vertical shape in case of buckling. This paper investigates the response of the rod deflection and the root mean square power output of selected vibration mode shapes with an attached tip mass.

Complex modal analysis and coupled electromechanical simulation of energy harvesting piezoelectric laminated beams

In this paper, coupled electromechanical behavior of a vibrational energy harvesting system composed of a unimorph piezoelectric laminated beam with a large attached tip mass is investigated. To achieve this goal, first the electromechanically coupled partial differential equations governing the lateral displacement and output voltage of the harvester are extracted through exploiting the Hamilton’s principle. Considering vibration damping due to mechanical to electrical energy conversion, a complex modal analysis is performed to extract the complex eigenfrequencies and eigenfunctions of the system. Furthermore, an exact analytical solution is presented for the system response to the harmonic base excitations, including output voltage and harvested power. To validate the analytical results, at the next step a finite element simulation is conducted through ABAQUS software. To perform a fully-coupled analysis which brings into account the effect of harvesting circuit, user subroutine User-defined Amplitude (UAMP) is utilized to calculate the voltage–current relation and impose the correct electrical charge on the electrodes in each step by monitoring the output voltage of the system at previous time increments. Results of both analytical and numerical simulations are compared for a Micro-Electro-Mechanical Systems (MEMS) harvester as a case study, where a very good agreement is observed between them.

Thermo-mechanical analysis of FG nanobeam with attached tip mass: an exact solution

Present disquisition proposes an analytical solution method for exploring the vibration characteristics of a cantilever functionally graded nanobeam with a concentrated mass exposed to thermal loading for the first time. Thermo-mechanical properties of FGM nanobeam are supposed to change through the thickness direction of beam based on the rule of power-law (P-FGM). The small-scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. Linear temperature rise (LTR) through thickness direction is studied. Existence of centralized mass in the free end of nanobeam influences the mechanical and physical properties. Timoshenko beam theory is employed to derive the nonlocal governing equations and boundary conditions of FGM beam attached with a tip mass under temperature field via Hamilton’s principle. An exact solution procedure is exploited to achieve the non-dimensional frequency of FG nanobeam exposed to temperature field with a tip mass. A parametric study is led to assess the efficacy of temperature changes, tip mass, small scale, beam thickness, power-law exponent, slenderness and thermal loading on the natural frequencies of FG cantilever nanobeam with a point mass at the free end. It is concluded that these parameters play remarkable roles on the dynamic behavior of FG nanobeam subjected to LTR with a tip mass. The results for simpler states are confirmed with known data in the literature. Presented numerical results can serve as benchmarks for future thermo-mechanical analyses of FG nanobeam with tip mass.

A practical method for calculating eigenfrequencies of a cantilever microbeam with the attached tip mass

This paper is concerned with the free vibration of cantilever microbeams with attached tip mass in a systematical manner. Small size effects on the vibrations of the microbeam are taken into consideration by introducing a scale parameter. A Fourier sine series is used to represent the lateral displacement function. Stokes’ transformation is applied in the present formulation and corresponding derivatives are presented explicitly. The present formulations can be readily reduced to those for others classical elasticity models by setting corresponding small scale parameter to zero. Several parametric studies are performed to validate the present solutions and the effects of various important physical parameters (scale parameter, tip mass) are investigated.

Design and performance analysis of a nanogyroscope based on electrostatic actuation and capacitive sensing

In this paper, a vibrating beam gyroscope with high operational frequencies at mode-matched condition is proposed. The model comprises a micro-cantilever with attached tip mass operating in the flextural–flextural mode. The drive mode is actuated via the electrostatic force, and due to the angular rotation of the base about the longitudinal axis. The secondary sub-nanometric vibration is induced in the sense direction which causes a capacitive change in the sense electrodes. The coupled electro-mechanical equation of motion is derived using the extended Hamilton's principle, and it is solved by direct numerical integration method. The gyroscope performance is investigated through the simulation results, where the device dynamic response, rate sensitivity, resolution, bandwidth, dynamic range, g sensitivity and shock resistance are studied. The obtained results show that the proposed device may have better performance compared to commercial micro electromechanical gyroscope characteristics.

Exact Frequency Analysis of a Rotating Cantilever Beam With Tip Mass Subjected to Torsional-Bending Vibrations

An exact frequency analysis of a rotating beam with an attached tip mass is addressed in this paper while the beam undergoes coupled torsional-bending vibrations. The governing coupled equations of motion and the corresponding boundary condition are derived in detail using the extended Hamilton principle. It has been shown that the source of coupling in the equations of motion is the rotation and that the equations are linked through the angular velocity of the base. Since the beam-tip-mass system at hand serves as the building block of many vibrating gyroscopic systems, which require high precision, a closed-form frequency equation of the system should be derived to determine its natural frequencies. The frequency analysis is the basis of the time domain analysis, and hence, the exact frequency derivation would lead to accurate time domain results, too. Control strategies of the aforementioned gyroscopic systems are mostly based on their resonant condition, and hence, acquiring knowledge about their exact natural frequencies could lead to a better control of the system. The parameter sensitivity analysis has been carried out to determine the effects of various system parameters on the natural frequencies. It has been shown that even the undamped systems undergoing base rotation will have complex eigenvalues, which demonstrate a damping-type behavior.

Study on Comparison of Atmospheric and Vacuum Environment of Thermally-Induced Vibration Using Vacuum Chamber

The present paper studies the thermally-induced vibration phenomenon of the flexible space boom structure. In order to simulate the thermally-induced vibration phenomenon of the flexible thin boom structure of the spacecraft with the attached tip mass in space, the thermally-induced vibration including thermal flutter is experimentally investigated at various thermal environments using a heating lamp in vacuum chamber. In this experimental study, fluctuating characteristics, natural frequency and thermal strains of the thermally-induced vibration are parametrically investigated at various thermal environment conditions. Finally the thermally-induced vibration of the flexible boom structure of the orbiting earth satellite in solar radiation environment from the earth eclipse region including umbra and penumbra is simulated using the power control of the heating lamp in the vacuum chamber.

유체유동을 갖는 외팔 파이프의 동특성 및 진동수에 미치는 설계인자의 영향

The vibrational system of this study consists of a cantilever pipe conveying fluid， the moving masses upon it and having an attached tip mass. The equation of motion is derived by using Lagrange's equation. The influences of the velocity and the inertia force of the moving mass and the velocities of fluid flow in the pipe have been studied on the dynamic behavior and the natural frequency of a cantilever pipe by numerical method. The deflection of the cantilever pipe conveying fluid is increased due to the tip mass and rotary inertia. Alter the moving mass passed upon the cantilever pipe, the amplitude of pipe is influenced by energy variation when the moving mass fall from the cantilever pipe. As the moving mass increase, the frequency of the cantilever pipe conveying fluid is increased. The rotary inertia of the tip mass influences much on the higher frequencies and vibration mode.

유체유동 외팔 파이프의 고유진동수에 미치는 이동질량들의 영향

The vibrational system of this study is consisted of a cantilever pipe conveying fluid, the moving masses upon it and an attached tip mass. The equation of motion is derived by using Lagrange equation. The influences of the velocity and the number of moving masses and the velocities of fluid flow in the pipe have been studied on the natural frequency of a cantilever pipe by numerical method. As the size and number of a moving mass increases, the natural frequency of cantilever pipe conveying fluid is decreased. When the first a moving mass Is located at the end of cantilever pipe, the increasing of the distance of moving masses make the natural frequency increase at first and third mode, but the frequency of second mode is decreased. The variation of natural frequency of the system is decreased due to increase of the number of a moving mass. The number and distance of moving masses effect more on the frequency of higher mode of vibration.

The Permanent Deformation of a Cracked Cantilever Struck Transversely at Its Tip

The presence of even a stable crack in a ductile cantilever can have a dramatic effect on the structural response of the beam. Not only can the magnitude of the permanent plastic deformation be significantly increased but also the final shape of the damaged beam can be dramatically affected by the size and location of the crack. Such effects are quantified by analyzing a simple model of the cracked beam with an attached tip mass.

Stability of a crack in a cantilever beam undergoing large plastic deformation after impact

A simple model is employed to determine the dynamic response of a rigid-perfectly plastic cantilever beam with an attached tip mass and a crack, taking into account the weakening effect of the crack. The crack is assumed to be located at the base of the beam, and an initial velocity is imparted to the tip mass. The subsequent stability of the crack is considered by calculating the tearing modulus based on the J- integral associated with the deflecting beam. For the example of circumferential cracks in thin-walled piping, whose idealized geometry models some stress corrosion cracks found in service, radial propagation and instability are found to be more likely than circumferential. Once a crack penetrates the wall, however, stability in the circumferential direction is found to depend in a complex way upon loading and crack geometry.

Natural frequencies of a cantilever with an asymmetrically attached tip mass

Pope, S. B. and Whitelaw, J. H., The Calculation of Near-Wake Journal of Fluid Mechanics, Vol. 73, Pt. 1, Jan. 1976, pp. 932. Lewelien, W. S., Teske, M., and Donaidson, C. Du P., Application of Turbulence Model Equations to Wakes, AIAA Journal, Vol. 12, May 1974, pp. 620-625. McGuirk, J. J. and Rodi, W., The Calculation of ThreeDirnensional Free Jets, Symposium on Turbulent Shear Flows, Pennsylvania State University, April 1977. Morse, A. P., Axisymmetric Turbulent Shear Flows with and without Swirl, Ph.D. Thesis, London University, England, 1977. 6 Rodi, W., The Prediction of Free Turbulent Boundary Layers by Use of a Two-Equation Model of Turbulence, Ph.D. thesis, London University, England, 1972. Pope, S. B., A Novel Calculation Procedure for Free Shear Fiows, Dept. of Mechanical Engineering, Imperial College, London, FS/77/8, 1977. Militzer, J., Nicholl, W. B., and Alpay, S. A., Some Observations on the Numerical Calculation of the Recirculation Region of Twin Parallel Symmetric Jet Symposium on Turbulent Shear Flows, Pennsylvania State University, April 1977.

Large Deformation of a Rigid, Ideally Plastic Cantilever Beam

A new formulation is presented for the deformation of a rigid, ideally plastic cantilever beam with an attached tip mass under impact loading at its tip. The effects due to arbitrarily large displacements are rigorously taken into account. The results are compared with Parkes’ solution2 in which the geometry changes are neglected. Comparisons with experiments3, 4 show that neglect of the geometry changes is responsible for part of the discrepancies between Parkes’ theory and the experiments.

The Plastic Deformation of a Cantilever Beam With Strain-Rate Sensitivity Under Impulsive Loading

This paper presents an analysis of the plastic deformation of a cantilever beam with an attached tip mass, based on the assumption of rigid-plastic behavior with strain-rate sensitivity, under an impulsive load at the tip. Numerical solutions of the equations of motion which appear as two nonlinear simultaneous integral equations are presented. The possibility of power-series expansion is also indicated. Finally, approximate solutions are given, and the results are compared with the more exact solutions and with experimental data.

The Plastic Deformation Due to Impact of a Cantilever Beam With an Attached Tip Mass

Abstract A theoretical analysis, based on the assumption of ideal plastic-rigid behavior, is made of the deformation of a uniform cantilever beam with an attached lip mass, which results from a transverse acceleration of the base of the cantilever. The problem is also investigated experimentally and an assessment is made of the theoretical work. A simplified analysis based on a one-dimensional model is found to give results which are very close to those obtained by analysis of the continuum model, but in both cases it is found that corrections must be applied to obtain good results, because of effects neglected in both analyses. In the case of the experimental models tested, the principal correction appears to be that for strain-rate effects.